Semigroups of matrices of intermediate growth

Finitely generated linear semigroups over a field K that have intermediate growth are considered. New classes of such semigroups are found and a conjecture on the equivalence of the subexponential growth of a finitely generated linear semigroup S and the nonexistence of free noncommutative subsemigroups in S, or equivalently the existence of a nontrivial identity satisfied in S, is stated. This ‘growth alternative’ conjecture is proved for linear semigroups of degree 2, 3 or 4. Certain results supporting the general conjecture are obtained. As the main tool, a new combinatorial property of groups is introduced and studied.     

Two crystals of doubly protonated melaminium salts: melaminium bis­(trifluoro­acetate) trihydrate and melaminium bis­(trichloro­acetate) dihydrate

Crystals of 2,4,6‐triamino‐1,3,5‐triazine‐1,3‐dium bis­(trifluoro­acetate) trihydrate, C₃H₈N₆²⁺·2CF₃COO⁻·3H₂O, and 2,4,6‐triamino‐1,3,5‐triazine‐1,3‐dium bis­(trichloro­acetate) dihydrate, C₃H₈N₆²⁺·2CCl₃COO⁻·2H₂O, both contain doubly protonated melamine rings that lie on crystallographic twofold axes. In the former structure, one water mol­ecule also lies on a twofold axis. While the trifluoro­acetate compound crystallizes in a centrosymmetric space group, the trichloro­acetate is non‐centrosymmetric, so it is useful as a material for non‐linear optics. The efficiency of second harmonic generation is about three times greater than that of KDP (KH₂PO₄). A combination of ionic and donor–acceptor hydrogen‐bond inter­actions link the melaminium(2+) residues with trifluoro­acetate or trichloro­acetate ions and water mol­ecules to form a three‐dimensional network.     

Risk management of a bond portfolio using options

In this paper, we elaborate a formula for determining the optimal strike price for a bond put option, used to hedge a position in a bond. This strike price is optimal in the sense that it minimizes, for a given budget, either Value-at-Risk or Tail Value-at-Risk. Formulas are derived for both zero-coupon and coupon bonds, which can also be understood as a portfolio of bonds. These formulas are valid for any short rate model that implies an affine term structure model and in particular that implies a lognormal distribution of future zero-coupon bond prices. As an application, we focus on the Hull-White one-factor model, which is calibrated to a set of cap prices. We illustrate our procedure by hedging a Belgian government bond, and take into account the possibility of divergence between theoretical option prices and real option prices. This paper can be seen as an extension of the work of Ahn and co-workers [Ahn, D., Boudoukh, J., Richardson, M., Whitelaw, R., 1999. Optimal risk management using options. J. Financ. 54, 359-375], who consider the same problem for an investment in a share.     

Photoreactions of benzonitrile with cyclic enol ethers

Upon irradiation, cyclic enol ethers such as 1-methoxy-cyclopentene (4) mainly add across the cyano group of benzonitrile (1), under formation of 2-azabutadienes of an imidoester type. This is in agreement with the so-called Δ G-correlation which was reported earlier (ref. 5 and 6). 4-Phenyloxazole (9) is formed from 1 and 1,3-dioxole (5) probably by a similar photochemical process followed by electrocyclic ring opening and hydrolysis. The low yield of the latter photoreaction and the almost exergonic electron transfer between 5 and 1 may point to back electron transfer as the main energy wasting process. From 1 and 2,3-dihydrido-furan (2) only the ortho cycloadduct 6 has been isolated in low yields.     

A Simple Bi-directional Mode Expansion Propagation Algorithm Based on Modes of a Parallel-plate Waveguide

The bi-directional mode expansion propagation algorithm (BEP) is known to be an accurate and efficient method for modelling field distribution in high-index contrast waveguide structures with strong back-reflections like Bragg gratings and photonic crystals. The main difficulty of this method is that for lossy structures, the propagation constants of modes are to be searched in the complex plane. To speed-up this procedure, a two-step algorithm for eigenmode calculation based on the expansion into the modes of an empty metallic waveguide has recently been proposed. Proper truncation rules possessing good convergence of the expansion method for both TE and TM modes have also been recently published. In this contribution, both these approaches are combined in the development of an extremely simple version of the two-dimensional BEP method that makes use of the field expansion into the eigenmodes of a parallel-plate waveguide. The method is strictly reciprocal and appeared to be computationally reliable also for strongly lossy structures. High numerical stability is ensured using the scattering matrix formalism, and an efficient method of calculating Bloch modes for symmetric as well as asymmetric periodic waveguide structures is adopted. A wide range of applicability of the method is demonstrated by a few examples.